Sharpness results concerning finite differences in Fourier analysis on the circle group
Let G denote the group R or T, let ι denote the identity element of G, and let s ∈ N be given. Then, a difference of order s is a function f ∈ L 2 (G) for which there are a ∈ G and g ∈ L 2 (G) such that f = (δι−δa) s ∗g. Let Ds(L 2 (G)) be the vector space of functions that are finite sums of differ...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2018
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| Sorozat: | Acta scientiarum mathematicarum
84 No. 3-4 |
| Kulcsszavak: | Fourier analízis |
| doi: | 10.14232/actasm-017-522-y |
| Online Access: | http://acta.bibl.u-szeged.hu/56930 |
| Tartalmi kivonat: | Let G denote the group R or T, let ι denote the identity element of G, and let s ∈ N be given. Then, a difference of order s is a function f ∈ L 2 (G) for which there are a ∈ G and g ∈ L 2 (G) such that f = (δι−δa) s ∗g. Let Ds(L 2 (G)) be the vector space of functions that are finite sums of differences of order s. It is known that if f ∈ L 2 (R), f ∈ Ds(L 2 (R)) if and only if R ∞ −∞ |fb(x)| 2 |x| −2s dx < ∞. Also, if f ∈ L 2 (T), f ∈ Ds(L 2 (T)) if and only if fb(0) = 0. Consequently, Ds(L 2 (G)) is a Hilbert space in a (possibly) weighted L 2 -norm. It is known that every function in Ds(L 2 (G)) is a sum of 2s + 1 differences of order s. However, there are functions in Ds(L 2 (R)) that are not a sum of 2s differences of order s, and we call the latter type of fact a sharpness result. In D1(L 2 (T)), it is known that there are functions that are not a sum of two differences of order one. A main aim here is to obtain new sharpness results in the spaces Ds(L 2 (T)) that complement the results known for R, but also to present new results in Ds(L 2 (T)) that do not correspond to known results in Ds(L 2 (R)). Some results are obtained using connections with Diophantine approximation. The techniques also use combinatorial estimates for potentials arising from points in the unit cube in Euclidean space, and make use of subtraction sets in arithmetic combinatorics. |
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| Terjedelem/Fizikai jellemzők: | 591-609 |
| ISSN: | 0001-6969 |